# Playing Impartial Games on a Simplicial Complex as an Extension of the Emperor Sum Theory

## Discrete and Computational Geometry, Graphs, and Games

## Keywords:

combinatorial game theory, impartial game, $\mathcal{P}$-position length, emperor sum## Abstract

In this paper, we considered impartial games on a simplicial complex. Each vertex of a given simplicial complex acts as a position of an impartial game. Each player in turn chooses a face of the simplicial complex and, for each position on each vertex of that face, the player can make an arbitrary number of moves. Moreover, the player can make only a single move for each position on each vertex, not on that face. We show how the $\mathcal{P}$-positions of this game can be characterized using the $\mathcal{P}$-position length. This result can be considered an extension of the emperor sum theory. While the emperor sum only allowed multiple moves for a single component, this study examines the case where multiple moves can be made for multiple components, and clarifies areas that the emperor sum theory did not cover.

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## How to Cite

*Thai Journal of Mathematics*,

*21*(4), 769–774. Retrieved from https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1543